432 XV NORMED ALGEBRAS AND SPECTRAL THEORY (15.13.7) The defect of a closed Hermitian operator H is defined to be the pair (m, ri) where m (resp. n) is equal to the dimension of E^ (resp. E#) if this dimension is finite, and is +00 otherwise; m is called the positive defect and n the negative defect of H. (15.13.8) For a closed hermitian operator H to be self-adjoint, it is necessary and sufficient that its defect should be (0, 0), or in other words that its Cayley transform V should be unitary. For H to be capable of extension to a self- adjoint operator, it is necessary and sufficient that its defect should be of the form (m, m). The first assertion follows trivially from (15.13.6). Next, if H is the re- striction of a self-adjoint operator A, then the Cayley transform V of H is the restriction of the Cayley transform UofA, which is unitary. Hence by defini- tion we have E# = J7(E#) because U is an automorphism of the Hilbert space E, and the defect of H is therefore of the form (m, m). Conversely, if this is so, then there exists an automorphism U of the Hilbert space E which coincides with V on dom(F) and is such that U(E£) = E# (because two Hilbert spaces of the same finite dimension, or two separable infinite- dimensional Hilbert spaces, are isomorphic (6.6.2)). Since U is a unitary operator on E, and the image of E under I— U (which contains dom(HJ) is dense in E, it follows that (7 is the Cayley transform of a self-adjoint operator which extends H. A hermitian operator H whose closure H** is self-adjoint is said to be essentially self-adjoint. Examples (15.13.9) Let (On^o be a Hilbert basis of an infinite-dimensional separ- able Hilbert space E. Let ¥ be the closed hyperplane in E generated by the en with n *z 1 (the orthogonal supplement of the line Ce0) and let U' be the isometry of F onto E such that U' • en = en.1 for all n ;> 1. Then the image G' of /— U' is dense in E. For G' contains the vectors en — en+1 for all n ^ 0, oo and if a vector x=£ £„ en in E is orthogonal to all the vectors em - em+± (m *> 0), w = 0 then we have £m = £m+1 for all m ^ 0, so that all the £„ (n ^ 0) are equal; and 00 this is not compatible with the convergence of the series £ |£J2 unless »=o) onto r(domCO), such that r= VR. (As in the proof of (15.12.8),